continuity and equicontinuity of semigroups on norming dual …...strongly continuous semigroups. on...

Semigroup Forum (2009) 79: 540–560DOI 10.1007/s00233-009-9174-9

R E S E A R C H A RT I C L E

Continuity and equicontinuity of semigroupson norming dual pairs

Markus Kunze

Received: 10 February 2009 / Accepted: 8 July 2009 / Published online: 1 August 2009© The Author(s) 2009. This article is published with open access at Springerlink.com

Abstract We study continuity and equicontinuity of semigroups on norming dualpairs with respect to topologies defined in terms of the duality. In particular, we ad-dress the question whether continuity of a semigroup already implies (local/quasi)equicontinuity. We apply our results to transition semigroups and show that, undersuitable hypothesis on E, every transition semigroup on Cb(E) which is continuouswith respect to the strict topology β0 is automatically quasi-equicontinuous with re-spect to that topology. We also give several characterizations of β0-continuous semi-groups on Cb(E) and provide a convenient condition for the transition semigroup ofa Banach space valued Markov process to be β0-continuous.

Keywords Norming dual pairs · Transition semigroups · Equicontinuity · Stricttopology

Introduction

An object of central interest in the study of Markov processes is the transition semi-group of the process. If the Markov process (Xt )t≥0 takes values in the measur-able space (E,�), the state space of the process, then the transition semigroupT = (T (t))t≥0 is a positive contraction semigroup on the space Bb(E) of all bounded,measurable functions on E. This semigroup contains all information about the tran-sition probabilities of Xt . More precisely, for t, s ≥ 0 and A ∈ � we have P(Xt+s ∈A|Xs = x) = (T (t)1A)(x).

Communicated by Rainer Nagel.

The author was partially supported by the Deutsche Forschungsgesellschaft in the framework of theDFG research training group 1100 at the University of Ulm.

M. Kunze (�)Delft Institute of Applied Mathematics, Delft University of Technology, P.O. Box 5031,2600 GA Delft, The Netherlandse-mail: [email protected]

mailto:[email protected]

Continuity and equicontinuity of semigroups 541

Whereas the orbits of the semigroup T usually bear no continuity properties, oftenthe restriction of T to certain invariant subspaces is continuous in one way or other.The best known example for this is that of a Feller semigroup. Here, E is a locallycompact Hausdorff space endowed with the Borel σ -algebra.

If C0(E), the space of all continuous functions on E which vanish at infinity, isinvariant under E, then often T|C0(E) is strongly continuous. This can be used to greateffect in the study of Markov processes, see [9, 15]. If E is not locally compact or ifC0(E) is not invariant, then one can consider other, invariant subspaces. Of particularinterest is the space Cb(E) of all bounded, continuous functions on E. However, evenif Cb(E) is invariant under T, the restriction of T to Cb(E) is in general not stronglycontinuous. But in many cases, see e.g. [7, 13, 14], the restriction is continuous withrespect to the so-called strict (or mixed) topology β0, cf. [3, 4, 24] and Sect. 1.3.

For locally compact spaces E, Sentilles [23] has studied β0-continuous semi-groups on Cb(E) in the framework of equicontinuous semigroups on locally con-vex spaces [25]. Since β0 agrees with the compact-open topology τco on ‖ · ‖∞-bounded subsets of Cb(E), it is also possible to treat β0-continuous semigroups asτco-continuous semigroups. This point of view was taken by Cerrai in [5] and led tothe concept of bi-continuous semigroups introduced by Kühnemund in [19]. Farkas[10] has used the theory of bi-continuous semigroups to study transition semigroupson Cb(E), where E is a Polish space, i.e. the topology of E is induced by a complete,separable metric. It should be noted that transition semigroups on Cb(E) in generaldo not satisfy the equicontinuity assumption of [25] with respect to τco, see [19, Ex-ample 6]. However, in the examples in [13, 14] local equicontinuity with respect toβ0 holds.

In this paper, we study continuity and equicontinuity of semigroups in the frame-work of semigroups on norming dual pairs introduced in [20]. Thus, in addition toa Banach space X, we are given a closed subspace Y of X∗, the norm dual of X,which is norming for X. We then study semigroups T on X such that the adjointsemigroup T∗ leaves the space Y invariant. In applications to transition semigroupswe will choose X = Cb(E), here the state space E is assumed to be a completelyregular Hausdorff space, and Y = M0(E), the space of all bounded Radon measureson E. In this context the assumption that T∗M0(E) ⊂ M0(E) is quite natural andhas a stochastic interpretation. Namely, if T is the transition semigroup of the Markovprocess (Xt )t≥0 and we put T′ = T∗|M0(E), then T′ gives the distribution of the ran-dom elements Xt , i.e. if Xs ∼ μ ∈ M0(E) then Xt+s ∼ T (t)′μ.

In Sects. 2 and 3 we will work on general norming dual pairs and study continuityand equicontinuity with respect to general locally convex topologies defined in termsof the duality, see Sect. 1.1. This generality allows us to consider continuity withrespect to various topologies. In particular, if we choose τ = ‖ · ‖, then we obtainstrongly continuous semigroups. On the norming dual pair (Cb(E), M0(E)) not onlythe strict topology β0 but also the weak topology σ(Cb(E), M0(E)) is of interest.This topology is connected with the concept of bounded and pointwise convergence,see [9, Sect. 3.4]. Priola [21] has used this continuity concept to study transitionsemigroups.

If we additionally impose certain equicontinuity assumptions, then it is not sur-prising that we can prove a generation theorem for such semigroups. The more in-teresting question is whether equicontinuity assumptions are restrictive or whether,

542 M. Kunze

at least for certain topologies, these assumptions are satisfied automatically. We willaddress this question in Sect. 3 and give some abstract examples where this is thecase. In Sect. 4 we apply our results to transition semigroups. We will prove that ifE is a Polish space, then every β0-continuous semigroup on (Cb(E), M0(E)) is lo-cally β0-equicontinuous. A variant of this result has been obtained independently byFarkas [10]. However, we will also prove that this result remains valid for positivesemigroups, whenever E is a so-called T-space (see the definition in Sect. 4). In themain result of Sect. 4, Theorem 4.4, we give various equivalent conditions for a semi-group on (Cb(E), M0(E)) to be β0-continuous. In the concluding Sect. 5 we discussseveral examples and give a convenient condition for the transition semigroup of aBanach space valued Markov process to be β0-continuous.

1 Preliminaries and notation

1.1 Dual pairs

Throughout this paper we will be working on dual pairs and use locally convextopologies defined in terms of the duality. We briefly recall some results from the the-ory and fix some notation. Our main reference are Chaps. 20 and 21 of [17]. A dualpair is a triple (X,Y, 〈 · , · 〉) where X and Y are vector spaces over the same fieldK = R or C and 〈 · , · 〉 is a bilinear form from X ×Y to K which separates points, i.e.〈x , y〉 = 0 for all x ∈ X implies y = 0 and 〈x , y〉 = 0 for all y ∈ Y implies x = 0.We may define locally convex topologies on X as follows. If M ⊂ Y is bounded,i.e. supy∈M |〈x , y〉| < ∞ for all x ∈ X, then pM(x) := supy∈M |〈x , y〉| defines aseminorm on X. If M is a collection of bounded subsets of Y , then the collection ofseminorms (pM)M∈M defines a locally convex topology on X if and only if for everyx ∈ X there exists some M ∈ M such that pM(x) �= 0 (we say that M is separating).If M is a separating collection of bounded subsets of Y , then τM denotes the locallyconvex topology induced by the seminorms (pM)M∈M.

A locally convex topology τ on X is called consistent if (X, τ)′ = Y , i.e. everyτ -continuous linear functional ϕ on X is of the form ϕ(x) = 〈x , y〉 for some y ∈ Y .By the Mackey-Arens theorem, [17, 21.4 (2)], every consistent topology is of theform τM for a suitable collection M. Furthermore, there exists a coarsest consistenttopology, namely the weak topology σ(X,Y ) = τF, where F denotes the collection ofall finite subsets of Y , and a finest consistent topology, namely the Mackey topologyμ(X,Y ) = τK, where K denotes the collection of all absolutely convex, σ(Y,X)-compact subsets of Y . We note that every topology τM is finer than the weak topologyσ(X,Y ). To simplify notation, we will write σ (resp. μ) for σ(X,Y ) (resp. μ(X,Y ))and denote σ -convergence on X by ⇀. We will write σ ′ (resp. μ′) for σ(Y,X) (resp.μ(Y,X)) and denote σ ′-convergence on Y by ⇀′.

1.2 Norming dual pairs

Definition 1.1 A norming dual pair is a dual pair (X,Y, 〈 · , · 〉) where X and Y

are Banach spaces and we have ‖x‖ = sup{|〈x , y〉| : y ∈ Y,‖y‖ ≤ 1} and ‖y‖ =sup{|〈x , y〉| : x ∈ X,‖x‖ ≤ 1}.


In what follows, we will often write (X,Y ) instead of (X,Y, 〈 · , · 〉) if the dualitypairing is understood. It is easy to see that if (X,Y ) is a norming dual pair, then Y

is isometrically isomorphic to a closed subspace of X∗, the norm dual of X. We willoften identify Y with this closed subspace of X∗.

It is an easy but crucial consequence of the definition that if (X,Y ) is a norm-ing dual pair, then on X and Y the notions of weak (i.e. σ - or σ ′-) boundednessand of norm boundedness coincide, cf. [20]. It follows that the norm topology onX is equal to τB, where B denotes the collection of all bounded subsets of Y . Thenorm topology is in general not consistent, but it is finer than any topology τM. It isproved in [20] that a σ -continuous linear operator is automatically ‖ · ‖-continuous.Furthermore, a ‖ · ‖-continuous linear operator T is σ -continuous if and only if itsnorm-adjoint T ∗ leaves the space Y invariant. By [17, 21.4 (6)], a linear operatoris σ -continuous if and only if it is μ-continuous. If τ is a consistent locally convextopology, then every τ -continuous linear operator is σ -continuous. The converse isnot true in general. If τ is any (not necessarily consistent) locally convex topologyon X, we write L(X, τ) for the algebra of τ -continuous linear operators on X. Forτ = ‖ · ‖ we merely write L(X) instead of L(X, ‖ · ‖). If T ∈ L(X,σ), we write T ∗for its norm adjoint and T ′ for its σ -adjoint. Note that T ′ = T ∗|Y .

1.3 The dual pair (Cb(E), M0(E))

Our main example for applications is the norming dual pair (Cb(E), M0(E)). Here,E is a completely regular Hausdorff space and Cb(E) denotes the Banach space ofall bounded, continuous functions form E to C endowed with the supremum norm.A positive measure μ, defined on the Borel σ -algebra B(E), is called a Radonmeasure if for all A ∈ B(E), we have μ(A) = sup{μ(K) : K ⊂ A,Kcompact}.If μ is a complex measure on B(E), then its total variation |μ| is defined by|μ|(A) = supZ

∑B∈Z |μ(B)|, where the supremum is taken over all finite partitions

Z of A into pairwise disjoint measurable sets. A complex measure μ is called a Radonmeasure, if |μ| is a Radon measure. Note that if E is a Polish space, then every mea-sure on B(E) is a Radon measure. M0(E) denotes the Banach space of all boundedRadon measures on E, endowed with the total variation norm ‖μ‖ := |μ|(E). Itis proved in [20] that (Cb(E), M0(E)) is a norming dual pair with respect tothe duality 〈f , μ〉 = ∫

Ef dμ. If T ∈ L(Cb(E),σ ), then T has the representation

Tf (x) = ∫E

f (y)k(x, dy). Here, k(x, ·) = T ′δx , where δx denotes the Dirac measurein x. We will call k the kernel associated with T . The question whether k(·,A) ismeasurable for all A ∈ B(E) is discussed in [20].

The strict topology β0 on Cb(E) is defined as follows:Denote by F0(E) the space of all bounded functions on E which vanish at infinity,

i.e. given ε > 0, we find a compact set K ⊂ E such that |f (x)| ≤ ε for all x �∈ K . Thestrict topology β0 on Cb(E) is the locally convex topology generated by the set ofseminorms (pϕ)ϕ∈F0(E), where pϕ(f ) := ‖ϕf ‖∞.

This definition is taken from [16]. It generalizes the definition given by Buck[3, 4] for locally compact spaces E. By [16, Theorem 7.6.3], (Cb(E),β0)

′ = M0(E),i.e. β0 is a consistent topology. Furthermore, (Cb(E),β0) is complete if and only ifC(E), the space of all continuous functions on E, is complete with respect to τco, see

544 M. Kunze

Theorems 4 and 9 in Sect. 3.6 of [16]. In particular, if E is a metric space or a locallycompact space, then (C0(E),β0) is complete. Sentilles [24] has considered severalstrict topologies yielding different spaces of measures as dual spaces. We will recallsome results from [24] in Sect. 4.

2 Semigroups and their generators

We now study semigroups on norming dual pairs. As a matter of fact, several in-teresting properties of such semigroups can be proved without continuity assump-tions, merely imposing integrability assumptions. This leads to the concept of inte-grable semigroups on norming dual pairs. Such semigroups are studied in [20] andwe content ourselves with recalling the definition and collecting some of the resultsfrom [20] in Propositions 2.3 and 2.4 below.

Definition 2.1 Let (X,Y ) be a norming dual pair. A semigroup on (X,Y ) is a familyT = (T (t))t≥0 ⊂ L(X,σ) such that

(1) T is a semigroup, i.e. T (0) = id and T (t + s) = T (t)T (s) for all t, s ≥ 0.(2) T is exponentially bounded, i.e. there exist M ≥ 1 and ω ∈ R such that ‖T (t)‖ ≤

Meωt for all t ≥ 0. We then say that T is of type (M,ω).

A semigroup T of type (M,ω) is called integrable if

(3) for all λ with Reλ > ω, there exists an operator R(λ) ∈ L(X,σ) such that

〈R(λ)x , y〉 =∫ ∞

0e−λt 〈T (t)x , y〉dt ∀x ∈ X,y ∈ Y. (2.1)

In particular, we assume that all the integrals on the right hand side exist. R =(R(λ))Reλ>ω is called the Laplace transform of T.

Remark 2.2 Even though in our definition of a semigroup on (X,Y ) only a semi-group on X appears, we are actually dealing with two semigroups simultaneously—a semigroup on X and a semigroup on Y . This resembles the situation for transitionsemigroups which serves as a leitmotif here.

Indeed since T ⊂ L(X,σ), it follows from the remarks in section 1.2 that T′ =(T (t)′)t≥0 ⊂ L(Y,σ ′). Since also R ⊂ L(X,σ), it follows that T is an integrablesemigroup on (X,Y ) if and only if T′ is an integrable semigroup on (Y,X). In thiscase, R′ is the Laplace transform of T′.

Proposition 2.3 Let T be an integrable semigroup of type (M,ω) with Laplace trans-form R.

(1) R is a pseudoresolvent and every R(λ) commutes with every T (t).(2) We have ‖(Reλ − ω)kR(λ)k‖ ≤ M for all Reλ > ω and k ∈ N.(3) If rgR is σ -dense in X, then R determines T uniquely, i.e. if T̃ is a second inte-

grable semigroup on (X,Y ) having the same Laplace transform R, then T = T̃.


Recall that a pseudoresolvent is a map R from some nonempty set � ⊂ C toL(X, ‖ · ‖), such that R(λ) − R(μ) = (μ − λ)R(λ)R(μ) for all λ,μ ∈ �. It is wellknown that for a given pseudoresolvent (R(λ))λ∈� there exists a unique multivaluedoperator A such that R(λ) = (λ− A)−1 for all λ ∈ �. In particular, the range rgR(λ)

and the kernel kerR(λ) are independent of λ ∈ �.The following proposition gives a characterization of the operator A. The integrals

appearing are to be understood as Y -integrals, see [20]. More precisely, if f : I → X

is a function defined on some interval I such that 〈f (·) , y〉 is integrable for everyy ∈ Y , then the Y -integral

∫If (t)dt denotes the unique element ϕ ∈ Y ∗ such that

ϕ(y) = ∫I〈f (t) , y〉dt for all y ∈ Y . In the proposition below,

∫If (t)dt will actu-

ally be an element of X which is considered as a closed subspace of Y ∗. Hence,∫ t

0 f (t)dt = x ∈ X if and only if 〈x , y〉 = ∫I〈f (t) , y〉dt for all y ∈ Y . However,

even if∫If (t)dt ∈ X, the integral in general does not exist as a Bochner or as a

Pettis integral.

Proposition 2.4 Let T be an integrable semigroup on the norming dual pair (X,Y )

with Laplace transform R(λ) = (λ − A)−1.

(1) The following are equivalent.(a) x ∈ D(A) and z ∈ Ax;(b) for every t > 0 and y ∈ Y we have

∫ t

0T (s)zds = T (t)x − x. (2.2)

(2) For x ∈ X and t > 0 we have∫ t

0 T (s)xds ∈ D(A) and

T (t)x − x ∈ A∫ t

0T (s)xds.

Remark 2.5 It follows from (2.2) that t �→ T (t)x is ‖ · ‖-continuous for everyx ∈ D(A). To see this, let x ∈ D(A) and z ∈ Ax be given. For every t0 > 0 wehave C := supt≤t0

‖T (t)‖ < ∞ and hence (2.2) implies that

|〈T (t)x − T (s)x , y〉| ≤∫ t

s

|〈T (r)z , y〉|dr ≤ |t − s|2C‖z‖ · ‖y‖

for t, s ≤ t0 and y ∈ Y . Taking the supremum over y ∈ Y with ‖y‖ ≤ 1, ‖ · ‖-continuity of t �→ T (t)x follows.

Definition 2.6 Let T be an integrable semigroup on the norming dual pair (X,Y )

such that the Laplace transform R of T is injective. Then the unique (single valued)operator A such that R(λ) = (λ − A)−1 is called the generator of T. In this case wesay that T has a generator or that T is a semigroup with generator A.

If τ is a locally convex topology on X, then, as usual, an operator A on X is calledτ -closed if the graph of A is closed in X×X with respect to τ ×τ . If τ is a consistent

546 M. Kunze

topology, then, by the Hahn-Banach theorem, an operator A is τ -closed if and only ifit is σ -closed. Furthermore, a σ -closed operator is automatically norm closed. For anoperator A we denote its resolvent set by ρ(A) and for λ ∈ ρ(A) we write R(λ,A)

for the resolvent of A in λ. We define

ρσ (A) := {λ ∈ ρ(A) : R(λ,A) ∈ L(X,σ)}.It is an open question whether ρσ (A) = ρ(A) for a σ -closed operator A. For a σ -densely defined, σ -closed operator, the σ -adjoint of A is denoted by A′.

Proposition 2.7 Let T be an integrable semigroup of type (M,ω) with generator A.Then A is a σ -closed operator with {Reλ > ω} ⊂ ρσ (A). Furthermore, for Reλ > ω

and k ∈ N0 we have

‖R(λ,A)k‖ ≤ M

(Reλ − ω)k. (2.3)

The operator A is σ -densely defined if and only if T′ has a generator.

Proof Since the resolvent of A is the Laplace transform of T and since the Laplacetransform consists of σ -continuous operators, {Reλ > ω} ⊂ ρσ (A). In particular, A

is σ -closed. Estimate (2.3) follows from Proposition 2.3. Now assume that A is σ -densely defined. In this case, the σ -adjoint A′ of A is well defined and R(λ,A)′ =R(λ,A′), as is easy to see. Since clearly

⟨x , R(λ,A)′y

⟩ = ∫ ∞0 e−λt

⟨x , T (t)′y

⟩dt for

all x ∈ X and y ∈ Y , it follows that A′ is the generator of T′. Conversely assume thatT′ has a generator B . As the Laplace transform of T′ is R(λ,A)′, we find R(λ,A)′ =R(λ,B). If y ∈ Y vanishes on D(A), then 0 = 〈R(λ,A)x , y〉 = 〈x , R(λ,B)y〉 forall x ∈ X. It follows that R(λ,B)y = 0. Since R(λ,B) is injective by hypothesis,y = 0 follows. By the Hahn-Banach theorem, D(A) is σ -dense in X. �

We now turn to continuous semigroups.

Definition 2.8 Let T be a semigroup on (X,Y ) and τ be a locally convex topologyon X. Then T is called τ -continuous (at 0) if for all x ∈ X the map t �→ T (t)x isτ -continuous (at 0).

Remark 2.9

(1) Using the uniform boundedness principle, it can be shown that if T is a semigroupon (X,Y ) which is σ -continuous at 0, then T is automatically exponentiallybounded, i.e. condition (2) in Definition 2.1 is automatically satisfied, see [20].

(2) In the remainder of this section, we will assume integrability of semigroups, i.e.condition (3) in Definition 2.1, also under continuity assumptions. This is dueto the fact that σ -continuity at 0 in general does not imply integrability of asemigroup, see the example in Sect. 5.2. However, it is proved in [20] that if E

is a complete metric space, then every semigroup on (Cb(E), M0(E)) which isσ -continuous at 0 is integrable.


Our definition of the generator via the Laplace transform is in the spirit of [1].The following theorem shows that, under continuity assumptions, this “integral defi-nition” coincides with the “differential definition” of the generator, see e.g. [8].

Theorem 2.10 Let T be an integrable semigroup on (X,Y ) of type (M,ω) and M

be a separating collection of bounded subsets of Y . If T is τM-continuous at 0, thenτM - limλ→∞ λR(λ)x = x. In particular, the Laplace transform of T is injective andT has a generator A such that D(A) is sequentially τM-dense in X. Furthermore,the generator is given by

D(A) ={

x ∈ X : τM - limh↓0

�hx exists

}

, Ax = τM - limh↓0

�hx,

where �hx := h−1(T (h)x − x).

Proof Let x ∈ X, S ∈ M and ε > 0 be given. Since S is bounded, there exists C > 0such that ‖y‖ ≤ C for all y ∈ S. By τM-continuity at 0, there exists t0 > 0 such that|〈T (t)x − x , y〉| ≤ ε for all t ≤ t0 and y ∈ S. Now for λ > max{ω,0} and y ∈ S wehave

supy∈S

|〈λR(λ)x − x , y〉| = supy∈S

∣∣∣∣

∫ ∞

0

⟨λe−λtT (t)x − λe−λtx , y

⟩dy

∣∣∣∣

≤ supy∈S

∫ t0

0λe−λt |〈T (t)x − x , y〉|dt

+∫ ∞

t0

λe−λt (1 + Meωt )C · ‖x‖dt

≤ ε(1 − e−λt0

) + C · ‖x‖(

e−λt0 + λ · Mλ − ω

e(ω−λ)t0

)

→ ε

as λ → ∞. Since S ∈ M was arbitrary, the first part is proved.Now denote the generator of T (in the sense of Definition 2.6) by B and let A be

the operator in the statement. If x ∈ D(B), then, by Proposition 2.4, we have

|〈�hx − Bx , y〉| ≤ 1

h

∫ h

0|〈T (s)Bx − Bx , y〉|ds,

for every y ∈ Y . Now let S ∈ M and ε > 0 be given. Choose t0 > 0 such thatpS(T (s)Bx − Bx) ≤ ε, for all 0 ≤ s ≤ t0. Then,

pS (�hx − Bx) ≤ 1

h

∫ h

0εds = ε,

for all 0 ≤ s ≤ t0. This proves that x ∈ D(A) and that Ax = Bx. Conversely supposethat x ∈ D(A). Since τM is finer than σ it follows that �hx ⇀ Ax as h ↓ 0. Since

548 M. Kunze

every operator T (s) is σ -continuous, T (s)�hx ⇀ T (s)Ax for every s ≥ 0. Further-more, (�hx)h≤1 is norm bounded. Indeed, for every y ∈ Y , the set {〈�hx , y〉 : h ≤ 1}is bounded. Hence, by the uniform boundedness principle, suph≤1 ‖�hx‖Y ∗ < ∞.However, since X embeds isometrically into Y ∗, we have ‖�hx‖Y ∗ = ‖�hx‖X forevery h > 0.

Now fix t > 0 and y ∈ Y . Put It := ∫ t

0 T (s)xds. Then

∫ t

0〈T (s)Ax , y〉ds = lim

h↓0

∫ t

0〈T (s)�hx , y〉ds = lim

h↓0〈�hIt , y〉,

by the dominated convergence theorem. Note that It ∈ D(B) and 〈BIt , y〉 =〈T (t)x − x , y〉 by Proposition 2.4. Since B ⊂ A, it follows that

∫ t

0〈T (s)Ax , y〉 = lim

h↓0〈�It , y〉 = 〈BIt , y〉 = 〈T (t)x − x , y〉.

Thus Proposition 2.4 implies that x ∈ D(B) and Bx = Ax. �

Remark 2.11 Assume in addition to the hypothesis of Theorem 2.10 that the semi-group T is τM-continuous. Then, arguing similar as in the proof of Theorem 2.10, itis easy to see that x ∈ D(A) if and only if t �→ T (t)x is τM-differentiable. In this casewe have d

dtT (t)x = T (t)Ax. Note however that τM-continuity at 0 does not imply

τM-continuity.

We now give a characterization of continuous semigroups.

Proposition 2.12 Let (X,Y ) be a norming dual pair and M be a separating col-lection of bounded subsets of Y . Furthermore, let T be an integrable semigroup on(X,Y ) with generator. Then the following are equivalent:

(1) T is τM-continuous.(2) For all t0 > 0 and every x ∈ X the set {T (t)x : t ∈ [0, t0]} is τM-compact.(3) For some t0 > 0 and every x ∈ X the set {T (t)x : t ∈ [0, t0]} is relatively count-

ably τM-compact.

Proof (1) ⇒ (2) and (2) ⇒ (3) are trivial. For (3) ⇒ (1) suppose that t �→ T (t)x isnot τM-continuous at t ∈ [0, t0]. Then there exists a τM-continuous seminorm p, anε > 0 and a sequence (tn) ⊂ [0, t0] converging to t such that p(T (tn)x − T (t)x) ≥ ε

for all n ∈ N. By hypothesis, the sequence T (tn)x has an accumulation point z ∈ X.

Thus there exists a subnet tα of tn such that T (tα)xτM→ z. Since τM is finer than σ we

have T (tα)x ⇀ z. As R(λ) is σ -continuous and commutes with the semigroup, wehave

R(λ)z = σ - limR(λ)T (tα)x = σ - limT (tα)R(λ)x = T (t)R(λ)x = R(λ)T (t)x,

since s �→ T (s)R(λ)x is ‖ · ‖-continuous. As R(λ) is injective, it follows that z =T (t)x. But then p(T (tα)x −T (t)x) → 0, a contradiction. This proves that t �→ T (t)x

is τM-continuous on [0, t0]. Using the semigroup law and that {T (t)T (t0)x : t ∈


[0, t0]} is relatively countably compact, it follows that t �→ T (t)x is τM-continuouson [0,2t0]. Inductively we obtain continuity for all times. �

3 Equicontinuity

In the context of semigroups, several types of equicontinuity assumptions have beendiscussed in the literature. We briefly recall the definitions.

Definition 3.1 Let (X, τ) be a locally convex space. A set S ⊂ L(X, τ) is calledequicontinuous, if for every τ -continuous seminorm p, there exists a τ -continuousseminorm q such that p(T x) ≤ q(x) for all x ∈ X and T ∈ S . A semigroup T ofτ -continuous operators is called locally τ -equicontinuous, if {T (t) : t ∈ [0, t0]} is τ -equicontinuous for all t0 > 0. It is called (globally) τ -equicontinuous, if {T (t) : t ≥ 0}is τ -equicontinuous. If for some α ∈ R the rescaled semigroup Tα := (e−αtT (t))t≥0is τ -equicontinuous, then T is called αquasi-τ -equicontinuous. We will say that T isquasi-τ -equicontinuous, if it is αquasi-τ -equicontinuous for some α ∈ R.

Obviously, every quasi-τ -equicontinuous semigroup (in particular, every τ -equicontinuous semigroup) is locally equicontinuous. The converse is not true ingeneral.

Example 3.2 Consider the norming dual pair (Cb(R), M0(R)). The compact opentopology τco is of the form τM. More precisely, M is the separating collection ofsets of the form {δx : x ∈ K}, where δx denotes the Dirac measure in x and K isa compact subset of R. The shift semigroup T, defined by T (t)f (x) = f (x + t) islocally τco-equicontinuous but not quasi-τco-equicontinuous.

Proposition 3.3 Let T be an integrable semigroup on (X,Y ) with generator A

and M be a separating collection of bounded subsets of Y . If T is locally τM-equicontinuous and D(A) is τM-dense in X, then T is τM-continuous.

Proof We first prove that X0 := {x ∈ X : t �→ T (t)x is τM - continuous} is τM-closed in X. Let x be an accumulation point of X0, t0 ≥ 0 and p be a τM-continuousseminorm. Pick a τM-continuous seminorm q such that p(T (t)z) ≤ q(z) for allt ∈ [0, t0 +1] and z ∈ X. Given ε > 0, we find x0 ∈ X0 such that q(x −x0) ≤ ε. Sincex0 ∈ X0, there exists 0 < δ < 1 such that p(T (t0)x0 −T (t)x0) ≤ ε for all |t − t0| ≤ δ.Now

p(T (t0)x − T (t)x) ≤ p(T (t0)x − T (t0)x0) + p(T (t0)x0 − T (t)x0)

+ p(T (t)x0 − T (t)x)

≤ 2q(x − x0) + p(T (t0)x0 − T (t)x0) ≤ 3ε,

for all |t − t0| ≤ δ. This proves that x ∈ X0 whence X0 is τM-closed. For x ∈ D(A),t �→ T (t)x is ‖ · ‖-continuous and hence τM-continuous. Thus D(A) ⊂ X0. As D(A)

is τM-dense, X0 = X follows. �

550 M. Kunze

For τM = ‖ · ‖, we note that local norm-equicontinuity of a semigroup is equiv-alent with exponential boundedness. Hence from Theorem 2.10 and Proposition 3.3we immediately obtain the following characterization.

Corollary 3.4 Let T be an integrable semigroup on (X,Y ). The following are equiv-alent:

(1) T is strongly continuous;(2) T has a ‖ · ‖-densely defined generator.

For quasi-equicontinuous semigroups, we obtain the following generation result.

Theorem 3.5 Let (X,Y ) be a norming dual pair, τ be a consistent topology on X

which is sequentially complete, and A be a σ -closed operator on X. The followingare equivalent.

(1) A is the generator of a τ -continuous, αquasi-τ -equicontinuous, integrable semi-group T on (X,Y ) of type (M,ω);

(2) A is a sequentially τ -densely defined operator such that(a) {λ ∈ R : λ > ω} ⊂ ρσ (A) and

‖(λ − ω)kR(λ,A)k‖ ≤ M ∀λ > ω,k ∈ N.

(b) The set{(λ − α)kR(λ,A)k : λ > α,k ∈ N

}

is τ -equicontinuous.

Proof (1) ⇒ (2): A is sequentially τ -densely defined by Theorem 2.10. Condi-tion (2)(a) follows directly from Proposition 2.7. As a resolvent, R(λ,A) satisfiesdk

dλk R(λ,A) = (−1)kk!R(λ,A)k+1. Interchanging differentiation and integration inthe formula for the Laplace transform yields

⟨R(λ,A)kx , y

⟩ = 1

(k − 1)!∫ ∞

0tk−1〈T (t)x , y〉dt, (3.1)

for all x ∈ X and y ∈ Y . Now let p be a τ -continuous seminorm. Since {e−αtT (t) :t ≥ 0} is τ -equicontinuous, we find a τ -continuous seminorm q such thatp(e−αtT (t)x) ≤ q(x) for all t ≥ 0 and x ∈ X. Since τ is consistent, it follows thatτ = τM for a suitable separating collection of bounded subsets of Y . We may thusassume that p = pS and q = pR for certain S,R ∈ M. For y ∈ S, k ∈ N and λ > α

we obtain from (3.1)

∣∣⟨(λ − α)kR(λ,A)kx , y

⟩∣∣ ≤ (λ − α)k

(k − 1)!∫ ∞

0tk−1e(α−λ)t

∣∣⟨eαtT (t)x , y

⟩∣∣dt

≤ (λ − α)k

(k − 1)!∫ ∞

0tk−1e(α−λ)t q(x)dt

= q(x).


Taking the supremum over y ∈ S, (2)(b) follows.(2) ⇒ (1) Let B = A − α. Since τ is sequentially complete, it follows from

(2)(b) and the theorem in Sect. IX.7 of [25] that B generates a τ -equicontinuous, τ -continuous semigroup S on X. Since τ is a consistent topology, L(X, τ) ⊂ L(X,σ)

and hence S is a semigroup on (X,Y ) (note that S is exponentially bounded by Re-mark 2.9(1)). Furthermore,

R(λ,B) = R -∫ ∞

0e−λtS(t)xdt,

where R -∫ ∞

0 denotes the improper Riemannian integral with respect to τ . How-ever, since the map x �→ 〈x , y〉 is τ -continuous for every y ∈ Y , it follows that〈R(λ,B)x , y〉 = ∫ ∞

0 e−λt 〈S(t)x , y〉dt for all x ∈ X and y ∈ Y . Thus, S is an inte-grable semigroup on (X,Y ) with generator B . Now put T (t) = eαtS(t). It is routineto check that T is an integrable semigroup with generator A. It remains to prove thatT is of type (M,ω). To that end, consider the rescaled semigroup Tω. Note that thegenerator of Tω is A − ω =: C. Now fix x ∈ X and y ∈ Y . The function ϕx,y : t �→〈e−ωtT (t)x, y〉 has Laplace transform 〈R(λ,C)x , y〉. Since ϕx,y is continuous, everypoint t ≥ 0 is a Lebesgue point of ϕx,y and we infer from the Post-Widder inversion

formula [1, Theorem 1.7.7] and the equation dk

dλk R(λ,C) = (−1)kk!R(λ,C)k+1 that

ϕx,y(t) = limn→∞

⟨[n

tR

(n

t,C

)]n

x, y

⟩

∀t ≥ 0.

However, since ‖λnR(λ,C)n‖ ≤ M , it follows that |〈e−ωtT (t)x, y〉| ≤ M‖x‖ · ‖y‖.Since Y is norming for X, it follows that T has type (M,ω). �

Remark 3.6

(1) The assumption that τ is sequentially complete and consistent is not needed inthe implication (1) ⇒ (2) in Theorem 3.5. In the implication (2) ⇒ (1), thesequential completeness is needed to apply the Theorem from [25], whereas theconsistency was used to ensure that L(X, τ) ⊂ L(X,σ).

(2) The proof of Theorem 3.4 shows that if τ is sequentially complete and consistent,then a τ -continuous, quasi-τ -equicontinuous semigroup is integrable.

The question remains whether quasi-equicontinuity is a mere technical assumptionin order to prove a Hille-Yosida type theorem or whether there are interesting casesin which continuity in a certain topology already implies quasi-equicontinuity. In[18] it is proved that on a barreled locally convex space (X, τ) (recall that (X, τ)

is called barreled if every absorbing, absolutely convex and closed subset of X is aτ -neighborhood of zero) every τ -continuous semigroup is locally τ -equicontinuous.The following proposition shows that on a norming dual pair consistent topologiesare never barreled, except when the norm topology is consistent. The special case(X,Y ) = (Cb(E), M0(E)) was considered in [24, Theorem 4.8].

Proposition 3.7 Let (X,Y ) be a norming dual pair and τ be a consistent topologyon X. The following are equivalent.

552 M. Kunze

(1) (X, τ) is barreled;(2) τ = ‖ · ‖ and thus Y = X∗.

Proof (1) ⇒ (2) If (X, τ) is barreled, then every weakly bounded subset of Y =(X, τ)′ is relatively σ ′-compact and τ = μ, see [17, 21.4 (4)]. However, if everyweakly bounded subset of Y is relatively σ ′-compact, then ‖ · ‖ = sup{y:‖y‖≤1} |〈· , y〉|is a μ-continuous seminorm and hence μ = ‖ · ‖.

(2) ⇒ (1) Is clear, since every normed space is barreled. �

However also in our general setting there are interesting examples in which conti-nuity with respect to τM of a semigroup on (X,Y ) implies quasi-τM-equicontinuity.We begin with the following

Lemma 3.8 Let (X,Y ) be a norming dual pair and M be a separating collection ofσ ′-compact subsets of Y . Further, let � be a compact Hausdorff space and F : � →L(X,σ) be strongly τM-continuous. Then for every K ∈ M the set

L K := {F(t)′y : t ∈ �,y ∈ K}

is σ ′-compact. If for every such L K there exists a set K0 ∈ M such that L K ⊂ K0,then {F(t) : t ∈ �} is τM-equicontinuous.

Proof We fix K ∈ M and write for simplicity L instead of L K . Let a net zα =F(tα)′yα ∈ L be given. Since � is compact, there exists a subnet tβ and some t0such that tβ → t0 in �. Since K is compact, there is a subnet yγ of yβ and an elementy0 ∈ K such that yγ ⇀′ y0. We claim that zγ = F(tγ )yγ ⇀′ z0 := F(t0)y0. To seethis, let x ∈ X be given. Then

∣∣⟨x , zγ − z0

⟩∣∣ ≤ ∣

∣⟨F(tγ )x − F(t0)x , yγ

⟩∣∣ + ∣

∣⟨F(t0)x , yγ − y0

⟩∣∣

≤ pK(F (tγ )x − F(t0)x) + ∣∣⟨F(t0)x , yγ − y0

⟩∣∣ → 0

by the τM-continuity of F(·)x and the weak convergence of yγ . This shows that L isσ ′-compact. We now prove the addendum. If L ⊂ K0 ∈ M, then

pK(F (t)x) = supy∈K

∣∣⟨x , F (t)′y

⟩∣∣ ≤ sup

y∈L|〈x , y〉| ≤ pK0(x)

for all x ∈ X and t ∈ �. Hence, if for every K ∈ M we find a set K0 ∈ M such thatthe above holds, it follows that F(�) is τM-equicontinuous. �

We immediately obtain:

Theorem 3.9 Let (X,Y ) be a norming dual pair and let τc := τC, where C is thecollection of all σ ′-compact subsets of Y . If T is a semigroup of type (M,ω) whichis τc-continuous, then T is αquasi-τc-equicontinuous for every α > ω.


Proof For α > ω and x ∈ X we have e−αtT (t)x → 0 in norm and hence with respectto the coarser topology τc. It follows that the map

F : [0,∞] → L(X,σ), F (t) ={

e−αtT (t), t ∈ [0,∞),

0, t = ∞is strongly τc-continuous. Now the statement follows from Lemma 3.8. �

We note that the topology τc is in general not consistent. However, it can happenthat the Mackey topology μ coincides with this topology ([6, 24] then say μ is thestrong Mackey topology of the pair (X,Y )). This is the case if and only if for everyσ ′-compact subset K of Y also its σ ′-closed, absolutely convex hull aco K is σ ′-compact. By Krein’s theorem [17, 24.5 (4)], if K is σ ′-compact then aco K is σ ′-compact if and only if aco K is μ′-complete. In particular, if μ′ is quasicomplete, i.e.μ′ is complete on every bounded, μ′-closed subset of X, then every μ-continuoussemigroup on X is quasi-μ-equicontinuous.

Corollary 3.10 If (X,Y ) is a norming dual pair such that μ′ is quasicomplete, thenevery μ-continuous semigroup T on (X,Y ) is quasi-μ-equicontinuous. In particular

(1) If T is a norm continuous semigroup on a Banach space X, then its adjoint semi-group T∗ on X∗ is μ(X∗,X)-continuous if and only if it is quasi-μ(X∗,X)-equicontinuous.

(2) If E is a completely regular Hausdorff space such that (Cb(E),β0) is complete,then every μ(M0(E),Cb(E))-continuous integrable semigroup on (M0(E),

Cb(E)) is quasi-μ(M0(E),Cb(E))-equicontinuous.

Proof The proof of the general statement was explained above. For (1) we note thatμ′ = μ(X,X∗) = ‖ · ‖ is complete whence every μ(X∗,X)-continuous adjoint semi-group is quasi-μ(X∗,X)-equicontinuous. The converse follows from Proposition 3.3since adjoint semigroups have a σ(X∗,X)-densely defined generator and hence, bythe Hahn-Banach theorem, a μ(X∗,X)-densely defined generator. For (2) observethat, as a consequence of Grothendieck’s completeness theorem [17, 21.9 (4)], theMackey topology μ(Cb(E), M0(E)) is complete, since there exists a complete, con-sistent topology, namely β0, on Cb(E). �

We will now apply Lemma 3.8 in the context of positive semigroups. We introducethe following notation. An ordered norming dual pair is a norming dual pair (X,Y )

where X is an ordered Banach space with σ -closed positive cone X+. Note that inthis case the dual cone Y+ := {y ∈ Y : 〈x , y〉 ≥ 0 ∀x ∈ X+} is σ ′-closed. As usual,we call T ∈ L(X,σ) positive if T X+ ⊂ X+. Note that in this case also T ′Y+ ⊂ Y+.

Theorem 3.11 Let (X,Y ) be an ordered norming dual pair and τ+ be the topologyof uniform convergence on the σ ′-compact subsets of Y+. If T is a positive, τ+-continuous semigroup of type (M,ω) on (X,Y ), then T is αquasi- τ+-equicontinuousfor every α > ω.

554 M. Kunze

Proof This follows from Lemma 3.8 as in the proof of Theorem 3.9, noting that forα > ω and K ⊂ Y+ the set {e−αtT (t)′y : t ≥ 0, y ∈ K} is not only compact but also asubset of Y+ by the positivity of the operators T (t). �

4 Applications to transition semigroups

In this section, we apply the results of the previous sections to semigroups on thenorming dual pair (Cb(E), M0(E)). Here, and throughout this section, E will be acompletely regular Hausdorff space. The consistent topology we are interested in isthe strict topology β0. In order to apply our results, we need additional informationabout β0 and the dual pair (Cb(E), M0(E)).

It is well known, see [17, 21.3 (2)], that if (X, τ) is any locally convex space thenτ is the topology of uniform convergence on the τ -equicontinuous subsets of (X, τ)′.For the strict topology, we have the following description of the β0-equicontinuoussubsets of M0(E).

Theorem 4.1 (Sentilles [24, Theorem 5.1]) A set H ⊂ M0(E) is β0-equicontinuousif and only if (1) supμ∈H |μ|(E) < ∞ and (2) for every ε > 0 there exists a compactset K ⊂ E such that |μ|(E \ K) ≤ ε for all μ ∈ H.

Condition (2) says that H is a tight set of measures. From Theorem 4.1 we inferthe following description of β0-equicontinuous sets of linear operators.

Proposition 4.2 Let S = {Tα : α ∈ I } ⊂ L(Cb(E),σ ) be a bounded family of opera-tors on Cb(E) with associated kernels pα . The following are equivalent.

(1) S is β0-equicontinuous;(2) given a compact subset K ⊂ E and ε > 0, there exists a compact subset L of E

such that

|pα|(x,E \ L) ≤ ε ∀x ∈ K,α ∈ I.

Proof (1) ⇒ (2): Let K ⊂ E be compact. Then K := {δx : x ∈ K} is β0-equicontin-uous by Theorem 4.1. In particular, pK is a β0-continuous seminorm. Since S isβ0-equicontinuous, we find a β0-continuous seminorm q such that pK(Tαf ) ≤ q(f )

for all f ∈ Cb(E) and α ∈ I . Note that q = pL for some β0-equicontinuous set L.We may assume that L is σ ′-closed and absolutely convex. But then it follows thatpα(x, ·) = T ′

αδx ∈ L for all x ∈ K . Indeed, if T ′α0

δx0 /∈ L for some α0 ∈ I and x0 ∈ K

then, as a consequence of the Hahn-Banach theorem, we can strictly separate T ′α0

δx0

from L, i.e. we find f ∈ Cb(E) = (M0(E),σ ′)′ and ε > 0 such that |〈f , μ〉| + ε ≤|〈f,T ′

α0δx0〉| for all μ ∈ L. But then pK(Tα0f ) ≥ pL(f ) + ε, a contradiction to the

choice of L. Hence, the set {pα(x, ·) : α ∈ I, x ∈ K} is a subset of L and thus β0-equicontinuous. Theorem 4.1 yields (2).

(2) ⇒ (1): Let H be a β0-equicontinuous subset of M0(E). Then there existsC > 0 such that ‖μ‖ = |μ|(E) ≤ C for all μ ∈ H. If we choose M > 0 such that‖Tα‖ ≤ M for all α ∈ I , then

∣∣⟨f , T ′

αμ⟩∣∣ ≤ M · C · ‖f ‖ ∀f ∈ Cb(E),α ∈ I,μ ∈ H.


Taking the supremum over f with ‖f ‖∞ ≤ 1, it follows that |T ′αμ|(E) ≤ C · M <

∞ for all α ∈ I and μ ∈ H. Furthermore, given ε > 0, we find a compact set K

such that |μ|(E \ K) ≤ ε2M

for all μ ∈ H. By (2), we find L ⊂ E compact such that|pα|(x,E \ L) ≤ ε

2Cfor all α ∈ I and x ∈ K . It follows that for μ ∈ H and α ∈ I we

have

|T ′αμ|(E \ L) ≤

∫

K

|pα|(x,E \ L)d|μ|(x) +∫

Kc

|pα|(x,E \ L)|d|μ|(x)

≤ ε

2C‖μ‖ + ε

2MM = ε.

Hence L := {T ′αμ : α ∈ I,μ ∈ H} is β0-equicontinuous and thus pL is a β0-

continuous seminorm. However, pH(Tαf ) ≤ pL(f ) for all f ∈ Cb(E). Since H wasarbitrary, it follows that S is β0-equicontinuous. �

Let us recall the following definition from [24]. A completely regular space E iscalled a T-space if every σ ′-compact set of positive Radon measures is tight. Thecelebrated Prokhorov theorem, see [22], asserts that every Polish space is a T-space.More generally, every complete metric space and every locally compact space is a T-space, see [24, Theorem 5.4]. If E is an infinite dimensional separable Hilbert spaceendowed with the weak topology, then E is not a T-space, cf. [12, Example I.6.4].

Theorem 4.3 Let τ+ denote the topology of uniform convergence on the σ ′-compactsubsets of M+

0 (E).

(1) β0 = τ+ iff E is a T -space.(2) If E is a T -space and every measure on E is a Radon measure, then β0 =

μ(Cb(E), M0(E)). In this case, every σ ′-compact subset of M0(E) is tight, i.e.β0 is the topology of uniform convergence on the σ ′-compact subsets of M0(E).

Proof (1) is [24, Theorem 5.3], (2) follows from Theorems 5.8 and 4.5 of that pa-per. �

We note that Conway [6] has proved that if E = [0,ω1), where ω1 is the firstuncountable ordinal and E is endowed with the order topology, then β0 is not theMackey topology of the pair (Cb(E), M0(E)). However, also in this case β0 = τ+since E, being locally compact, is a T-space.

We now come to the main result of this section.

Theorem 4.4 Let E be a T -space and let T be an integrable semigroup on(Cb(E), M0(E)). If there exists a measure μ on B(E) which is not a Radon measure,then additionally assume that T is positive. We denote the kernel associated to T (t)

by pt . Consider the following statements

(1) For every f ∈ Cb(E), the map (t, x) �→ T (t)f (x) is continuous;(2) For every f ∈ Cb(E) we have T (t)f → T (s)f for t → s uniformly on the com-

pact subsets of E;(3) T is a β0-continuous semigroup;

556 M. Kunze

(4) T is a quasi-β0-equicontinuous, β0-continuous semigroup;(5) T has a σ -densely defined generator and, given a compact subset K ⊂ E and

constants t0, ε > 0 there exists a compact subset L ⊂ E such that

|pt |(x,E \ L) ≤ ε ∀x ∈ K, t ∈ [0, t0].

Then (1) ⇒ (2) ⇔ (3) ⇔ (4) ⇔ (5). If E is either locally compact or a metric space,all five statements are equivalent.

Proof (1) ⇒ (2): Fix f ∈ Cb(E) and s ≥ 0. By assumption, for every ε > 0 andx ∈ E we find δ = δ(s, x) and a neighborhood U = U(s, x) of x such that

|T (s)f (x) − T (t)f (y)| < ε ∀(t, y) ∈ B(s, δ) × U.

Now let K ⊂ E compact. Then {s}×K ⊂ ⋃x∈K B(s, δ(s, x))×U(s, x). As {s}×K

is compact in [0,∞)×E we find finitely many x1, . . . , xn and δi := δ(s, xi) such that{s}×K ⊂ ⋃n

i=1 B(s, δi)×U(xi). Put δ = min{δ1, . . . , δn}. For x ∈ K , there exists i0such that x ∈ U(xi0). For |t − s| < δ we have

|T (t)f (x)−T (s)f (x)| ≤ |T (t)f (x)−T (s)f (xi0)|+|T (s)f (xi0)−T (s)f (x)| < 2ε,

since (t, x), (s, x) ∈ B(s, δi0) × U(xi0). As x ∈ K was arbitrary, we havesupx∈K |T (t)f (x) − T (s)f (x)| ≤ 2ε for |t − s| < δ. This proves (2).

(2) ⇒ (1) If E is a metric space, then (t, x) �→ T (t)f (x) is continuous iff it is se-quentially continuous. So let (tn, xn) → (s, x0). By (2), T (tn)f → T (s)f uniformlyon the compact set K = {xn : n ∈ N0}. But then T (tn)f (xn) → T (s)f (x0) followsusing the continuity of T (s)f .

Now assume that E is locally compact. Fix (s, x0) ∈ [0,∞) × E and f ∈ Cb(E).Since T (s)f is continuous, given ε > 0, there is a neighborhood U(x0) such that|T (s)f (x) − T (s)f (x0)| < ε for all x ∈ U(x0). It is no restriction to assume thatU(x0) is relatively compact. By (2) we find δ > 0 such that |T (t)f (x)−T (s)f (x)| <ε for all x ∈ U(x0) and all |t − s| < δ. Thus |T (t)f (x) − T (s)f (x0)| < 2ε for all(t, x) ∈ B(s, δ) × U(x0). This proves (1).

(2) ⇔ (3): Is clear since T is locally bounded and since the strict topology coin-cides with the compact-open topology on norm-bounded sets.

(3) ⇔ (4): If every measure on E is a Radon measure, then, by Theorem 4.3 (2),β0 is the topology of uniform convergence on the σ ′-compact subsets of M0(E) and(3) ⇒ (4) follows from Theorem 3.9. In the other case, β0 is the topology of uniformconvergence on the σ ′-compact subsets of M0(E)+ by Theorem 4.3 (1). Thus T isquasi-β0-equicontinuous as a consequence of the positivity of T and Theorem 3.11.This shows (3) ⇒ (4), the converse implication is trivial.

(4) ⇒ (5): Follows from Theorem 2.10 and Proposition 4.2.(5) ⇒ (4): Is a consequence of Propositions 4.2 and 3.3. �

Remark 4.5 The assumption in Theorem 4.4 that T is an integrable semigroup is onlyneeded for the equivalence (4) ⇔ (5).


Theorem 4.4 can be used to establish that a given transition semigroup on Cb(E)

is β0-continuous. In [2], transition semigroups are constructed from the solutions ofparabolic partial differential equations. Here E is a subset of R

d . For such transi-tion semigroups, condition (1) can easily be verified, as the PDE techniques yieldsolutions of the PDE which are continuous in both time and space variables. If onefollows [5, 11, 19] and prefers to think about semigroups on Cb(E) which have τco-continuous orbits, then of course Condition (2) is satisfied. In the next section, wewill show that if T is the transition semigroup of a Markov process obtained fromsolving a stochastic differential equation, then Condition (5) can often be verified.

We note that the crucial assumption in Theorem 4.4 is that T consists ofσ(Cb(E), M0(E))-continuous operators. Under suitable assumption on E, e.g. ifE is a Polish space, an operator T on Cb(E) is σ(Cb(E), M0(E))-continuous ifand only if it is a kernel operator, see [20]. If one follows the approach of [2], thenit is a consequence of the PDE techniques that the operators of the semigroup arerepresented by a Green function and thus are kernel operators.

If E is a Polish space and T is a τco-bi-continuous semigroup on Cb(E),then it follows from the definition of a bi-continuous semigroup that every oper-ator T (t) is sequentially τco-continuous on normbounded sets and hence sequen-tially β0-continuous. By [24, Corollary 8.4] it follows that T (t) ∈ L(Cb(E),β0) ⊂L(Cb(E),σ ). Farkas [10, Example 3.9] has given an example of a τco-bi-continuoussemigroup which does not consist of β0-continuous operators and is thus, in par-ticular, not locally β0-equicontinuous. In that example, E = [0,ω1) with the ordertopology, where ω1 is the first uncountable ordinal. Note that since [0,ω1) is locallycompact and hence a T-space, it follows from Theorem 4.4 that every positive β0-continuous semigroup of operators in L(Cb(E),σ ) is quasi-β0-equicontinuous.

5 Examples

5.1 The case E = N

If E = N endowed with the discrete topology, then Cb(E) = �∞ andM0(E) = �1. Thus in this case, M0(E) is the predual of Cb(E). The weak topol-ogy σ = σ(Cb(E), M0(E)) is merely the weak∗-topology of �∞ whereas the weaktopology σ ′ = σ(M0(E),Cb(E)) is the weak topology (in the Banach space sense)of �1. A bounded operator T on �∞ is σ -continuous if and only if it is the adjoint ofa bounded operator on �1. We now have the following result.

Proposition 5.1 If E = N endowed with the discrete topology, then every semigroupT on (Cb(E), M0(E)) which is σ -continuous at 0 is β0-continuous and quasi-β0-equicontinuous.

Proof If T is a semigroup on (�∞, �1) then T = S∗ for some semigroup S on �1. NowT is σ(�∞, �1)-continuous (at 0) if and only if S is σ(�1, �∞)-continuous (at 0). Itis well known (but also follows from Theorem 2.10 and Corollary 3.4) that a semi-group of bounded operators on a Banach space X which is weakly continuous at 0

558 M. Kunze

is already ‖ · ‖-continuous. It follows that T is σ(�∞, �1)-continuous. In particular,t �→ T (t)f (n) is continuous for every n ∈ N and every f ∈ Cb(N). However, sinceevery compact subset of N is finite, it follows that t �→ T (t)f is τco-continuous andhence β0-continuous for every f ∈ Cb(N). Since N is locally compact and since everymeasure on N is a Radon measure, the assertion follows from Theorem 4.4. �

5.2 The Sorgenfrey line

Let us consider the real line R endowed with the Sorgenfrey topology τs , i.e. τs

is generated by the intervals of the form [a, b). It follows that the Borel σ -algebraof (R, τs) is the usual Borel σ -algebra of R (with the metric | · |). It is wellknown that every compact subset of (R, τs) is countable. However, as the exam-ple {1 − 1

n: n ∈ N} shows, not every countable subset of R is compact in (R, τs).

It follows that every Radon measure on (R, τs) is concentrated on a countable set.Hence M0(R, τs) = �1(R), the space of all discrete measures on R. A function f onR is continuous with respect to τs if and only if it is right-continuous. Thus in thissituation (Cb(R, τs), M0(R, τs)) = (Cr(R), �1(R)), where Cr(R) denotes the spaceof all bounded, right continuous functions on R.

Consider the shift semigroup T given by T (t)f (x) = f (x + t). Then T is a posi-tive contraction semigroup on (Cr(R), �1(R)). However, T is not integrable. Indeed,it is easy to see that the Laplace transform of T is given by

R(λ)f (x) = eλx

∫ ∞

x

e−λyf (y)dy.

But this operator is not σ -continuous since R(λ)∗δ0 = e−λt1(0,∞)dt /∈ �1(R). Fur-thermore, T is σ -continuous at 0 but not σ -continuous. Since a continuous functionis uniformly continuous on compact sets, it follows that T (t)f

τco→ f as t ↓ 0 for everyf ∈ Cr(R). Hence T is β0-continuous at 0.

5.3 Applications to Markov processes

If E is a Banach space, then some Markov processes are obtained as solutions ofstochastic differential equations, see e.g. [13, 14]. The transition semigroup of sucha Markov process is defined as follows. If X(t, x) denotes the solution of the sto-chastic differential equation with initial datum x ∈ E, then one defines T (t)f (x) =E(f (X(t, x))). It is natural to ask for a condition for T to be β0-continuous in termsof properties of the map (t, x) �→ X(t, x). In applications, it frequently happens thatX(t, x) ∈ Lp(�,E), where � is the underlying probability space and 1 ≤ p < ∞,and the map (t, x) �→ X(t, x) is continuous. The following theorem shows that in thiscase the semigroup T is indeed β0-continuous. However, this result remains true in aeven more general setting.

If (�,F ,P ) is a probability space and (E,ρ) is a complete metric space, thenL0(�,E) denotes the space of all strongly measurable maps X : � → E (moduloequality P -almost everywhere) endowed with the topology of convergence in mea-sure.


Theorem 5.2 Let (�,F ,P ) be a probability space, (E,ρ) be a complete metricspace and X : [0,∞) × E → L0(�,E) be a continuous map. Define T (t)f (x) =E(f (X(t, x)) for f ∈ Cb(E). Then, for every t0 > 0, the set {T (t) : 0 ≤ t ≤ t0} is aβ0-equicontinuous family of operators on (Cb(E), M0(E)). If (T (t))t≥0 is a semi-group, then it is β0-continuous and quasi-β0-equicontinuous.

Proof Consider the map � : L0(�,E) → M0(E) given by �(X) = μX , where μX

denotes the distribution of X. Note that for X ∈ L0(�,E) the distribution μX isindeed a Radon measure since X has separable range. The map � is continuous.Indeed, if Xn → X in measure then, passing to a subsequence, we have Xn → X

almost everywhere. But then

⟨f , μXn

⟩ =∫

�

f (Xn)dP →∫

�

f (X)dP = 〈f , μX〉.

Thus, every subsequence of �(Xn) has a subsequence which converges to �(X) withrespect to σ(M0(E),Cb(E)).

Since T (t)f (x) = ⟨f , μX(t,x)

⟩, the continuity of x �→ X(t, x) for fixed t implies

that every operator T (t) maps Cb(E) into Cb(E). It follows from the joint continuityof X(·, ·) that for every t0 > 0 and every compact set K ⊂ E the set {X(t, x) : 0 ≤t ≤ t0, x ∈ K} is compact in L0(�,E). Hence the set {μX(t,x) : 0 ≤ t ≤ t0, x ∈ K}is σ ′-compact in M0(E)+. Since E is a complete metric space and hence a T-space, it follows that the latter set is tight. Hence Proposition 4.2 implies that theset {T (t) : 0 ≤ t ≤ t0} is β0-equicontinuous. In particular, every single operator T (t)

is β0-continuous and hence an element of L(Cb(E),σ ). Now assume that (T (t))t≥0is a semigroup. Since t �→ X(t, x) is continuous, it follows that t �→ T (t)f (x) is con-tinuous for every x ∈ E and hence (T (t))t≥0 is σ -continuous. Taking Remark 2.9 intoaccount, it follows from Proposition 2.10 that T has a σ -densely defined generator.The claim follows from Theorem 4.4. �

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